Let \(G\subset \operatorname{SL}_n(\)  be a finite subgroup. When \(n=2\)  there is a famous ADE classification of such subgroups that matches the classification of Du Val or modality zero singularities by taking a subgroup \(G\)  to the quotient singularity \(0\in \mathbb {A}^2/G\) . This observation and the surrounding deep interactions of the geometry of \(\mathbb {A}^2/G\)  and its resolutions, and the representation theory of \(G\)  are known as the two-dimensional McKay correspondence [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. In this case, the unique minimal or crepant resolution has a modular interpretation as the \(G\) -Hilbert scheme \(G\text{\textnormal {-Hilb}}\,\mathbb {A}^2\) . The moduli space \(G\text{\textnormal {-Hilb}}\,M\)  for \(M\)  a variety and \(G\subset \operatorname{Aut}(M)\)  a finite subgroup parameterises \(G\) -clusters in \(M\) : zero-dimensional \(G\) -invariant subschemes \(M\)  of \(\mathbb {A}^2\)  with \(H^0(\mathcal {O}_Z)\cong G]\)  as \(G\) -modules. This was generalised to three dimensions for finite abelian subgroups of \(\operatorname{SL}_3(\)  by Nakamura [8]} who showed that \(G\text{\textnormal {-Hilb}}\,\mathbb {A}^3\)  is a crepant resolution of \(\mathbb {A}^3/G\)  and then to all subgroups \(G\)  by the celebrated work of Bridgeland–King–Reid [9]}. They moreover established an equivalence of categories
\(  D^b\big (G\text{\textnormal {-Hilb}}\,\mathbb {A}^3\big )\simeq D^b_G\big (\mathbb {A}^3\big ),\) 
